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minimization_of_state_machines [2008/09/20 18:07] vkuncak |
minimization_of_state_machines [2015/04/21 17:32] (current) |
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This is the process of //minimization// of $M$. | This is the process of //minimization// of $M$. | ||
+ | * an easy case of minimizing size of 'generated code' in compiler | ||
We say that state machine $M$ distinguishes strings $w$ and $w'$ iff it is not the case that ($w \in L(M)$ iff $w' \in L(M)$). | We say that state machine $M$ distinguishes strings $w$ and $w'$ iff it is not the case that ($w \in L(M)$ iff $w' \in L(M)$). | ||
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We wish to merge states $q$ and $q'$ into same group as long as they "behave the same" on all future strings $w$, i.e. | We wish to merge states $q$ and $q'$ into same group as long as they "behave the same" on all future strings $w$, i.e. | ||
- | \[ | + | \begin{equation*} |
\delta(q,w) \in F \mbox{ iff } \delta(q',w) \in F \ \ \ (*) | \delta(q,w) \in F \mbox{ iff } \delta(q',w) \in F \ \ \ (*) | ||
- | \] | + | \end{equation*} |
for all $w$. | for all $w$. | ||
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- initially put $\nu = (Q \cap F) \times (Q \setminus F)$ (only final and non-final states are initially non-equivalent) | - initially put $\nu = (Q \cap F) \times (Q \setminus F)$ (only final and non-final states are initially non-equivalent) | ||
- repeat until no more changes: if $(r,r') \notin \nu$ and there is $a \in \Sigma$ such that $(\delta(r,a),\delta(r',a)) \in \nu$, then | - repeat until no more changes: if $(r,r') \notin \nu$ and there is $a \in \Sigma$ such that $(\delta(r,a),\delta(r',a)) \in \nu$, then | ||
- | \[ | + | \begin{equation*} |
\nu := \nu \cup \{(r,r')\} | \nu := \nu \cup \{(r,r')\} | ||
- | \] | + | \end{equation*} |
=== Step 3: Merge States that are not non-equivalent === | === Step 3: Merge States that are not non-equivalent === | ||
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Consequently, $Q^2 \setminus \nu$ is the equivalence relation. From the definition of this equivalence it follows that if two states are equivalent, then so is the result of applying $\delta$ to them. Therefore, we have obtained a well-defined deterministic automaton. | Consequently, $Q^2 \setminus \nu$ is the equivalence relation. From the definition of this equivalence it follows that if two states are equivalent, then so is the result of applying $\delta$ to them. Therefore, we have obtained a well-defined deterministic automaton. | ||
+ | |||
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Note that if two distinct states are non-equivalent, there is $w$ such that states $\delta(q_0,s_q w)$ and $\delta(q_0,s_{q'} w)$ have different acceptance, so $M$ distinguishes $s_q w$ and $s_{q'}w$. Now, if we take any other state machine $M' = (\Sigma,Q',\delta',q'_0,F')$ with $L(M')=L(M)$, it means that $\delta'(q'_0,s_q) \neq \delta'(q'_0,s_{q'})$, otherwise $M'$ would not distinguish $s_q w$ and $s_{q'} w$. So, if there are $K$ pairwise non-equivalent states in $M$, then a minimal finite state machine for $L(M)$ must have at least $K$ states. Note that if the algorithm constructs a state machine with $K$ states, it means that $Q^2 \setminus \tau$ had $K$ equivalence relations, which means that there exist $K$ non-equivalent states. Therefore, any other deterministic machine will have at least $K$ states, proving that the constructed machine is minimal. | Note that if two distinct states are non-equivalent, there is $w$ such that states $\delta(q_0,s_q w)$ and $\delta(q_0,s_{q'} w)$ have different acceptance, so $M$ distinguishes $s_q w$ and $s_{q'}w$. Now, if we take any other state machine $M' = (\Sigma,Q',\delta',q'_0,F')$ with $L(M')=L(M)$, it means that $\delta'(q'_0,s_q) \neq \delta'(q'_0,s_{q'})$, otherwise $M'$ would not distinguish $s_q w$ and $s_{q'} w$. So, if there are $K$ pairwise non-equivalent states in $M$, then a minimal finite state machine for $L(M)$ must have at least $K$ states. Note that if the algorithm constructs a state machine with $K$ states, it means that $Q^2 \setminus \tau$ had $K$ equivalence relations, which means that there exist $K$ non-equivalent states. Therefore, any other deterministic machine will have at least $K$ states, proving that the constructed machine is minimal. | ||
+ | === Example === | ||
+ | |||
+ | Construct automaton recognizing | ||
+ | * language {=,<nowiki><=</nowiki>} | ||
+ | * language {=,<nowiki><=</nowiki>,<nowiki>==</nowiki>} | ||
+ | Minimize the automaton. |